A Convex Cone is a Convex Set

Last updated Nov 1, 2022

Let $V$ be a Vector Space on $\mathbb{R}$ and let $S \subset V$ be a Convex Cone. Then $S$ is a Convex Set.

Suppose $\mathbf{u}, \mathbf{v} \in S$ and $a \in [0, 1]$. If $a = 0$ or $a = 1$ we have that $a \mathbf{u} + (1-a) \mathbf{v} \in {u, v} \subset S$. Otherwise $a \in (0,1)$ so $a > 0$ and $1-a > 0$. Because $S$ is a Convex Cone, we have that $a \mathbf{u} + (1-a) \mathbf{v} \in S$. Thus $S$ is a Convex Set. $\blacksquare$